#8762027

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#58169	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4111 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,18,23,27)(19,22,28,24)(20,29,31,25)(21,26,32,30)(33,34)(35,37)(36,44)(38,42)(39,41)(40,46)(43,47)(45,48);(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(17,19,23,28)(18,24,27,22)(20,21,31,32)(25,26,29,30)(33,34,38,42)(35,37,41,39)(36,40,46,44)(43,45,48,47);(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(17,20,23,31)(18,25,27,29)(19,21,28,32)(22,30,24,26)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,20)(18,25)(19,21)(22,30)(23,31)(24,26)(27,29)(28,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,21)(18,26)(19,20)(22,29)(23,32)(24,25)(27,30)(28,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,22,23,24)(18,19,27,28)(20,30,31,26)(21,29,32,25)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669223);